Hi,
I see three potential issues there:
1. You have a javascript error on that page:
monosnap.com/file/hvOs4HxAtVBsaVkhpHJ3QksrbxASvb
That javascript error during the loading of the page could prevent recaptcha from initializing itself.
2. I see that your registration view file has been customized. Rename temporarily the folder templates/YOUR_TEMPLATE/html/com_hikashop/user/ via FTP in order to deactivate the view override. If your override is made on an old version of the view file, it could be missing some key element to display the captcha properly.
3. It could be an issue with the settings of the plugin. Could you provide a screenshot of the settings of the plugin (you can blur/hide the key settings) ?